Liberty University BUSI 230 week 3 exercises 4.1-4.3 complete solutions answers and more!

Liberty University BUSI 230 week 3 exercises 4.1-4.3 complete solutions answers and more!

Question 1

Suppose the newspaper states that the probability of rain today is 90%.

What is the complement of the event "rain today"?

What is the probability of the complement? (Enter your answer to two decimal places.)

Question 2

What is the probability of the following.

(a) An event A that is certain to occur?

(b) An event B that is impossible?

Question 3

What is the law of large numbers?

If you were using the relative frequency of an event to estimate the probability of the event, would it be better to use 100 trials or 500 trials? Explain.

Question 4

A recent survey of 1090 U.S. adults selected at random showed that 685 consider the occupation of firefighter to have very great prestige. Estimate the probability (to the nearest hundredth) that a U.S. adult selected at random thinks the occupation of firefighter has very great prestige.

Question 5

Consider a family with 4 children. Assume the probability that one child is a boy is 0.5 and the probability that one child is a girl is also 0.5, and that the events "boy" and "girl" are independent.

(a) List the equally likely events for the gender of the 4 children, from oldest to youngest. (Let M represent a boy (male) and F represent a girl (female). Select all that apply.)

(b) What is the probability that all 4 children are male? (Enter your answer as a fraction.)

Notice that the complement of the event "all four children are male" is "at least one of the children is female." Use this information to compute the probability that at least one child is female. (Enter your answer as a fraction.)

Question 6

Consider the following.

(a) Explain why −0.41 cannot be the probability of some event.

(b) Explain why 1.21 cannot be the probability of some event.

(c) Explain why 120% cannot be the probability of some event.

(d) Can the number 0.56 be the probability of an event? Explain.

Question 7

Isabel Briggs Myers was a pioneer in the study of personality types. The personality types are broadly defined according to four main preferences. Do married couples choose similar or different personality types in their mates? The following data give an indication.

Similarities and Differences in a Random Sample of 375 Married Couples

Suppose that a married couple is selected at random.

(a) Use the data to estimate the probability that they will have 0, 1, 2, 3, or 4 personality preferences in common. (Enter your answers to 2 decimal places.)

(b) Do the probabilities add up to 1? Why should they?

What is the sample space in this problem?

Question 8

(a) If you roll a single die and count the number of dots on top, what is the sample space of all possible outcomes? Are the outcomes equally likely?

(b) Assign probabilities to the outcomes of the sample space of part (a). (Enter your answers as fractions.)

Do the probabilities add up to 1? Should they add up to 1? Explain.

(c) What is the probability of getting a number less than 6 on a single throw? (Enter your answer as a fraction.)

(d) What is the probability of getting 1 or 2 on a single throw? (Enter your answer as a fraction.)

Question 9

A botanist has developed a new hybrid cotton plant that can withstand insects better than other cotton plants. However, there is some concern about the germination of seeds from the new plant. To estimate the probability that a seed from the new plant will germinate, a random sample of 3000 seeds was planted in warm, moist soil. Of these seeds, 2550 germinated.

(a) Use relative frequencies to estimate the probability that a seed will germinate. What is your estimate? (Enter your answer to 3 decimal places.)

(b) Use relative frequencies to estimate the probability that a seed will not germinate. What is your estimate? (Enter your answer to 3 decimal places.)

(c) Either a seed germinates or it does not. What is the sample space in this problem?

Do the probabilities assigned to the sample space add up to 1? Should they add up to 1? Explain.

(d) Are the outcomes in the sample space of part (c) equally likely?

Question 10

John runs a computer software store. Yesterday he counted 121 people who walked by the store, 62 of whom came into the store. Of the 62, only 25 bought something in the store. (Round your answers to two decimal places.)

(a) Estimate the probability that a person who walks by the store will enter the store.

(b) Estimate the probability that a person who walks into the store will buy something.

(c) Estimate the probability that a person who walks by the store will come in and buy something.

(d) Estimate the probability that a person who comes into the store will buy nothing.

Question 11

If two events are mutually exclusive, can they occur concurrently? Explain.

Question 12

If two events A and B are independent and you know that P(A) = 0.20, what is the value of P(A | B)?

Question 13

Given P(A) = 0.7 and P(B) = 0.2, do the following.

(a) If A and B are mutually exclusive events, compute
 P(A or B).

(b) If P(A and B) = 0.1, compute P(A or B).

Question 14

Given P(A) = 0.4 and P(B) = 0.8, do the following.

(a) If A and B are independent events, compute
 P(A and B).

(b) If P(A | B) = 0.5, compute P(A and B).

Question 15

Given P(A) = 0.2, P(B) = 0.7, P(A | B) = 0.3, do the following.

(a) Compute P(A and B).

(b) Compute P(A or B).

Question 16

Consider the following events for a driver selected at random from the general population.

A = driver is under 25 years old

B = driver has received a speeding ticket

Translate each of the following phrases into symbols.

(a) The probability the driver has received a speeding ticket and is under 25 years old.

(b) The probability a driver who is under 25 years old has recieved a speeding ticket.

(c) The probability a driver who has received a speeding ticket is 25 years old or older.

(d) The probability the driver is under 25 years old or has received a speeding ticket.

(e) The probability the driver has not received a speeding ticket or is under 25 years old.

Question 17

M&M plain candies come in various colors. According to the M&M/Mars Department of Consumer Affairs, the distribution of colors for plain M&M candies is as follows.

color

percentage

Suppose you have a large bag of plain M&M candies and you choose one candy at random.

(a) Find P(green candy or blue candy).

Are these outcomes mutually exclusive? Why?

(b) Find P(yellow candy or red candy).

Are these outcomes mutually exclusive? Why?

(c) Find P(not purple candy).

Question 18

A national park is famous for its beautiful desert landscape and its many natural rock formations. The following table is based on information gathered by a park ranger of all rock formations of at least 3 feet. The height of the rock formation is rounded to the nearest foot.

Height of rock formation, feet

Number of rock formations in park

For a rock formation chosen at random from this park, use the preceding information to estimate the probability that the height of the rock formation is as follows. (Round your answers to two decimal places.)

(a) 3 to 9 feet

(b) 30 feet or taller

(c) 3 to 49 feet

(d) 10 to 74 feet

(e) 75 feet or taller

Question 19

A recent study gave the information shown in the table about ages of children receiving toys. The percentages represent all toys sold.

Age (years)

2 and under

(3-5)

(6-9)

(10-12)

(13 and over)

What is the probability that a toy is purchased for someone in the following age ranges?

(a) 6 years old or older

(b) 12 years old or younger

(c) between 6 and 12 years old

(d) between 3 and 9 years old

A child between 10 and 12 years old looks at this probability distribution and asks, "Why are people more likely to buy toys for kids older than I am (13 and over) than for kids in my age group (10–12)?" How would you respond?

Question 20

In a sales effectiveness seminar, a group of sales representatives tried two approaches to selling a customer a new automobile: the aggressive approach and the passive approach. For 1160 customers, the following record was kept:

Suppose a customer is selected at random from the 1160 participating customers. Let us use the following notation for events: A = aggressive approach, Pa = passive approach, S = sale, N = no sale. So, P(A) is the probability that an aggressive approach was used, and so on.

(a) Compute P(S), P(S | A), and P(S | Pa). (Enter your answers as fractions.)

(b) Are the events S = sale and Pa = passive approach independent? Explain.

(c) Compute P(A and S) and P(Pa and S). (Enter your answers as fractions.)

(d) Compute P(N) and P(N | A). (Enter your answers as fractions.)

(e) Are the events N = no sale and A aggressive approach independent? Explain.

(f) Compute P(A or S). (Enter your answer as a fraction.)

Question 21

A particular shoe franchise knows that its stores will not show a profit unless they gross over $940,000 per year. Let A be the event that a new store grosses over $940,000 its first year. Let B be the event that a store grosses over $940,000 its second year. The franchise has an administrative policy of closing a new store if it does not show a profit in either of the first 2 years. The accounting office at the franchise provided the following information: 67% of all the franchise stores show a profit the first year; 71% of all the franchise stores show a profit the second year (this includes stores that did not show a profit the first year); however, 81% of the franchise stores that showed a profit the first year also showed a profit the second year. Compute the following. (Enter your answers to four decimal places.)

(a)    P(A)

Answer

(b)    P(B)

Answer

(c)    P(B | A)

Answer

(d)    P(A and B)

Answer

(e)    P(A or B)

Answer

(f)    What is the probability that a new store will not be closed after 2 years?

Answer

What is the probability that a new store will be closed after 2 years?

Answer

Question 22

For each of the following situations, explain why the combinations rule or the permutations rule should be used.

(a) Determine the number of different groups of 5 items that can be selected from 12 distinct items.

Answer

(b) Determine the number of different arrangements of 5 items that can be selected from 12 distinct items.

Answer

Question 23

Consider the following.

(a) Draw a tree diagram to display all the possible outcomes that can occur when you flip a coin and then toss a die.

(b) How many outcomes contain a head and a number greater than 4?

Answer

(c) Probability extension: Assuming the outcomes displayed in the tree diagram are all equally likely, what is the probability that you will get a head and a number greater than 4 when you flip a coin and toss a die? (Round your answer to three decimal places.)

Answer

Question 24

There are ten balls in an urn. They are identical except for color. Five are red, four are blue, and one is yellow. You are to draw a ball from the urn, note its color, and set it aside. Then you are to draw another ball from the urn and note its color.

(a) Make a tree diagram to show all possible outcomes of the experiment.

(b) Let P(x, y) be the probability of choosing an x-colored ball on the first draw and a y-colored ball on the second draw. Compute the probability for each outcome of the experiment. (Enter your answers as fractions.)

P(R, R) =

P(R, B) =

P(R, Y) = 

P(B, R) =

P(B, B) =

P(B, Y) =

P(Y, R) = 

P(Y, B) =

Question 25

There are seven wires which need to be attached to a circuit board. A robotic device will attach the wires. The wires can be attached in any order, and the production manager wishes to determine which order would be fastest for the robot to use. Use the multiplication rule of counting to determine the number of possible sequences of assembly that must be tested. (Hint: There are seven choices for the first wire, six for the second wire, five for the third wire, etc.)

Question 26

Barbara is a research biologist for Green Carpet Lawns. She is studying the effects of fertilizer type, temperature at time of application, and water treatment after application. She has six fertilizer types, five temperature zones, and five water treatments to test. Determine the number of different lawn plots she needs in order to test each fertilizer type, temperature range, and water treatment configuration.

Question 27

Compute P5,2.

Question 28

Compute C7,2.

Question 29

In the Cash Now lottery game there are 15 finalists who submitted entry tickets on time. From these 15 tickets, three grand prize winners will be drawn. The first prize is one million dollars, the second prize is one hundred thousand dollars, and the third prize is ten thousand dollars. Determine the total number of different ways in which the winners can be drawn. (Assume that the tickets are not replaced after they are drawn.)

Question 30

There are 12 qualified applicants for 5 trainee positions in a fast-food management program. How many different groups of trainees can be selected?